Postprocessed integrators for the high order integration of ergodic SDEs

The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a numerical method by using an appropriate change of variables called the processor. We show that this technique can be extended to the stochastic context for the construction of new high order integrators for the sampling of the invariant measure of ergodic systems. The approach is illustrated with modifications of the stochastic $\theta$-method applied to Brownian dynamics, where postprocessors achieving order two are introduced. Numerical experiments, including stiff ergodic systems, illustrate the efficiency and versatility of the approach.Comment: 21 pages, to appear in SIAM J. Sci. Compu

A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

Stochastic finite differences and multilevel Monte Carlo for a class of SPDEs in finance

In this article, we propose a Milstein finite difference scheme for a stochastic partial differential equation (SPDE) describing a large particle system. We show, by means of Fourier analysis, that the discretisation on an unbounded domain is convergent of first order in the timestep and second order in the spatial grid size, and that the discretisation is stable with respect to boundary data. Numerical experiments clearly indicate that the same convergence order also holds for boundary-value problems. Multilevel path simulation, previously used for SDEs, is shown to give substantial complexity gains compared to a standard discretisation of the SPDE or direct simulation of the particle system. We derive complexity bounds and illustrate the results by an application to basket credit derivatives

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients: applications to financial models

A novel class of implicit Milstein type methods is devised and analyzed in the present work for stochastic differential equations (SDEs) with non-globally Lipschitz drift and diffusion coefficients. By incorporating a pair of method parameters $\theta, \eta \in [0, 1]$ into both the drift and diffusion parts, the new schemes can be viewed as a kind of double implicit methods, which also work for non-commutative noise driven SDEs. Within a general framework, we offer upper mean-square error bounds for the proposed schemes, based on certain error terms only getting involved with the exact solution processes. Such error bounds help us to easily analyze mean-square convergence rates of the schemes, without relying on a priori high-order moment estimates of numerical approximations. Putting further globally polynomial growth condition, we successfully recover the expected mean-square convergence rate of order one for the considered schemes with $\theta \in [\tfrac12, 1]$, solving general SDEs in various circumstances. As applications, some of the proposed schemes are also applied to solve two scalar SDE models arising in mathematical finance and evolving in the positive domain $(0, \infty)$. More specifically, the particular drift-diffusion implicit Milstein method ($\theta = \eta = 1$) is utilized to approximate the Heston $\tfrac32$-volatility model and the semi-implicit Milstein method ($\theta =1, \eta = 0$) is used to solve the Ait-Sahalia interest rate model. With the aid of the previously obtained error bounds, we reveal a mean-square convergence rate of order one of the positivity preserving schemes for the first time under more relaxed conditions, compared with existing relevant results for first order schemes in the literature. Numerical examples are finally reported to confirm the previous findings.Comment: 36 pages, 3 figure

Numerical contractivity preserving implicit balanced Milstein-type schemes for SDEs with non-global Lipschitz coefficients

Stability analysis, which was investigated in this paper, is one of the main issues related to numerical analysis for stochastic dynamical systems (SDS) and has the same important significance as the convergence one. To this end, we introduced the concept of $p$-th moment stability for the $n$-dimensional nonlinear stochastic differential equations (SDEs). Specifically, if $p = 2$ and the $p$-th moment stability constant \bar{K} < 0 , we speak of strict mean square contractivity. The present paper put the emphasis on systematic analysis of the numerical mean square contractivity of two kinds of implicit balanced Milstein-type schemes, e.g., the drift implicit balanced Milstein (DIBM) scheme and the semi-implicit balanced Milstein (SIBM) scheme (or double-implicit balanced Milstein scheme), for SDEs with non-global Lipschitz coefficients. The requirement in this paper allowed the drift coefficient $f(x)$ to satisfy a one-sided Lipschitz condition, while the diffusion coefficient $g(x)$ and the diffusion function $L^{1}g(x)$ are globally Lipschitz continuous, which includes the well-known stochastic Ginzburg Landau equation as an example. It was proved that both of the mentioned schemes can well preserve the numerical counterpart of the mean square contractivity of the underlying SDEs under appropriate conditions. These outcomes indicate under what conditions initial perturbations are under control and, thus, have no significant impact on numerical dynamic behavior during the numerical integration process. Finally, numerical experiments intuitively illustrated the theoretical results